#!/usr/bin/env python3

import functools
import math
from typing import List
from heapq import heappush, heappop


class RunningMedian(object):
    """A running median computer.

    >>> median = RunningMedian()
    >>> median.add_number(1)
    >>> median.add_number(10)
    >>> median.add_number(3)
    >>> median.get_median()
    3
    >>> median.add_number(7)
    >>> median.add_number(5)
    >>> median.get_median()
    5
    """

    def __init__(self):
        self.lowers, self.highers = [], []

    def add_number(self, number):
        if not self.highers or number > self.highers[0]:
            heappush(self.highers, number)
        else:
            heappush(self.lowers, -number)  # for lowers we need a max heap
        self.rebalance()

    def rebalance(self):
        if len(self.lowers) - len(self.highers) > 1:
            heappush(self.highers, -heappop(self.lowers))
        elif len(self.highers) - len(self.lowers) > 1:
            heappush(self.lowers, -heappop(self.highers))

    def get_median(self):
        if len(self.lowers) == len(self.highers):
            return (-self.lowers[0] + self.highers[0]) / 2
        elif len(self.lowers) > len(self.highers):
            return -self.lowers[0]
        else:
            return self.highers[0]


def gcd_floats(a: float, b: float) -> float:
    if a < b:
        return gcd_floats(b, a)

    # base case
    if abs(b) < 0.001:
        return a
    return gcd_floats(b, a - math.floor(a / b) * b)


def gcd_float_sequence(lst: List[float]) -> float:
    if len(lst) <= 0:
        raise ValueError("Need at least one number")
    elif len(lst) == 1:
        return lst[0]
    assert len(lst) >= 2
    gcd = gcd_floats(lst[0], lst[1])
    for i in range(2, len(lst)):
        gcd = gcd_floats(gcd, lst[i])
    return gcd


def truncate_float(n: float, decimals: int = 2):
    """
    Truncate a float to a particular number of decimals.

    >>> truncate_float(3.1415927, 3)
    3.141

    """
    assert decimals > 0 and decimals < 10
    multiplier = 10 ** decimals
    return int(n * multiplier) / multiplier


def percentage_to_multiplier(percent: float) -> float:
    """Given a percentage (e.g. 155%), return a factor needed to scale a
    number by that percentage.

    >>> percentage_to_multiplier(155)
    2.55
    >>> percentage_to_multiplier(45)
    1.45
    >>> percentage_to_multiplier(-25)
    0.75

    """
    multiplier = percent / 100
    multiplier += 1.0
    return multiplier


def multiplier_to_percent(multiplier: float) -> float:
    """Convert a multiplicative factor into a percent change.

    >>> multiplier_to_percent(0.75)
    -25.0
    >>> multiplier_to_percent(1.0)
    0.0
    >>> multiplier_to_percent(1.99)
    99.0

    """
    percent = multiplier
    if percent > 0.0:
        percent -= 1.0
    else:
        percent = 1.0 - percent
    percent *= 100.0
    return percent


@functools.lru_cache(maxsize=1024, typed=True)
def is_prime(n: int) -> bool:
    """
    Returns True if n is prime and False otherwise.  Obviously(?) very slow for
    very large input numbers.

    >>> is_prime(13)
    True
    >>> is_prime(22)
    False
    >>> is_prime(51602981)
    True

    """
    if not isinstance(n, int):
        raise TypeError("argument passed to is_prime is not of 'int' type")

    # Corner cases
    if n <= 1:
        return False
    if n <= 3:
        return True

    # This is checked so that we can skip middle five numbers in below
    # loop
    if n % 2 == 0 or n % 3 == 0:
        return False

    i = 5
    while i * i <= n:
        if n % i == 0 or n % (i + 2) == 0:
            return False
        i = i + 6
    return True


if __name__ == '__main__':
    import doctest

    doctest.testmod()